Emmy Noether is described as the most important female mathematician, but she also made a profound contribution to theoretical physics. Her theorem on the fundamental relationship between symmetry and conservation principles is extremely simple:Normal | Teacher | Scholar
For any property of a physical system that is symmetric, there is a corresponding conservation law.Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example, if a physical system is symmetric under rotations, its angular momentum is conserved. If it is symmetric in time, its energy is conserved. If it is symmetric in space, its momentum is conserved. Note the connection between these symmetries and the various forms of Heisenberg uncertainty principle.
ΔJ Δφ ≥ ℏ ΔE Δt ≥ ℏ Δp Δx ≥ ℏJ and φ are the "action-angle" variables that played a critical role in the development of the quantum theory of atomic structure. A great deal of modern physics starts with symmetries and symmetry breaking.
Symmetry and Asymmetry in Information PhysicsWhen a physical system has a symmetry of some sort, Noether's theorem describes a generator of the (local) symmetry group. In the Standard Model of Particle Physics, a symmetry generator is described as a conserved current. The thing that "flows" in the current is called the "Noether charge." The word "charge" is used as a synonym for "generator" in referring to the generator of the (local) symmetry group. The most important asymmetry in information physics is the result of the irreversibility of information creation, a consequence of the second law of thermodynamics, which demands that positive entropy greater than the negative entropy (information) created locally, be carried away if the local information is to be stable - and thus observable and possibly become a measurement. This asymmetry of information creation depends on the cosmic asymmetry in the expansion of the universe. According to Noether's theorem, this asymmetry implies that information is not conserved (contrary to the opinions of many mathematical physicists and computer scientists). The history of cosmic evolution, biological evolution, and cultural evolution is at every level a story of irreversible information creation by cosmic, biological, and human creative forces. Information philosophy shoes that we are co-creators of our universe. A very important symmetry in information physics helps to explain the puzzle of entanglement, which was first described as a paradox in the 1935 Einstein-Podolsky-Rosen thought experiment. The collapse of a two-particle wave function is symmetric in space and synchronous in time, for a special frame in which the source of the entangled particles and their mean motions are at rest. Almost every presentation of the EPR paradox begins with something like "Alice observes one particle..." and concludes with the question "How does the second particle get the information needed so that Bob's later measurements correlate perfectly with Alice?" There is a fundamental asymmetry in this framing of the EPR experiment. It is a surprise that Einstein, who was so good at seeing deep symmetries, did not consider how to remove the asymmetry. See a symmetric reframing of the EPR paradox.
Noether Charges and the Standard Model of Particle PhysicsThe standard model introduces various charge quantum numbers. These are examples of Noether symmetry generators (or currents of Noether charges). They include:
Gauge FieldsIn the case of local, dynamical symmetries, associated with every charge (particle) is a gauge field (wave). When it is quantized, the gauge field becomes a gauge boson. The charges of the theory "radiate" the gauge field. The gauge field of electromagnetism is the electromagnetic field. The gauge boson is the photon. The gauge field is information about the possibilities of a gauge boson being in a particular location. The probabilistic gauge field moves deterministically. The motion of the gauge boson is probabilistic. More precisely, when the symmetry group is a Lie group, then the charges are understood to correspond to the root system of the Lie group; the discreteness of the root system accounting for the quantization of the charge.
Restoring Noether Symmetry to Entangled ParticlesConsider this reframing of Entanglement in the Einstein-Podolsky-Rosen Paradox Alice's measurement collapses the two-particle wave function. The two indistinguishable particles simultaneously appear at locations in a space-like separation. The frame of reference in which the source of the two entangled particles and the two experimenters are at rest is a special frame in the following sense. As Einstein knew very well, there are frames of reference moving with respect to the laboratory frame of the two observers in which the time order of the events can be reversed. In some moving frames Alice measures first, but in others Bob measures first. If there is a special frame of reference (not a preferred frame in the relativistic sense), surely it is the one in which the origin of the two entangled particles is at rest. Assuming that Alice and Bob are also at rest in this special frame and equidistant from the origin, we arrive at the simple picture in which any measurement that causes the two-particle wave function to collapse makes both particles appear simultaneously at determinate places with fully correlated properties (just those that are needed to conserve energy, momentum, angular momentum, and spin).